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Applying generalized maximum principle and weak maximum principle, we obtain several uniqueness results for spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker (GRW) space-time under suitable geometric assumptions. Furthermore, we also study the special case when the ambient space is static and provide some results by using Bochner’s formula.

In recent years, spacelike hypersurfaces in Lorentzian manifolds have been deeply studied not only from their mathematical interest, but also from their importance in general relativity.

Particularly, there are many articles that study spacelike hypersurfaces in weighted warped product space-times. A weighted manifold is a Riemannian manifold with a measure that has a smooth positive density with respect to the Riemannian one. More precisely, the weighted manifold

In [

In this paper we study spacelike hypersurfaces in a weighted

We have organized this paper as follows. In Section

Let

Recall that a smooth immersion

In the following, we will deal with two particular functions naturally attached to spacelike hypersurface

Let

Now, we consider that a GRW space-time

For a smooth function

According to Gromov [

It follows from a splitting theorem due to Case (see [

In the following, we give some technical lemmas that will be essential for the proofs of our main results in weighted GRW space-times

Let

If we denote

Let

In the following, we will introduce the weak maximum principle for the drifted Laplacian. By the fact in [

Let

In this section, we will state and prove our main results in weighted GRW space-times

Recall that a slab of a weighted GRW spacetime

Let

From (

By the hypotheses, we have

Let

By a similar reasoning as in the proof of Theorem

Taking into account the assumptions, we have

Next, we will use the weak maximum principle to study the rigidity of the spacelike hypersurfaces in weighted GRW space-times.

Let

We take the Gauss map

By Lemma

On the other hand, from (

Considering that the function

In this section, we obtain some rigidity results of stochastically complete hypersurfaces in weighted static GRW space-times

Let

Let

Let

By a direct computation and considering the hypothesis

Furthermore, taking into account that the weighted function

Therefore,

Now we recall the Bochner-Lichnerowicz formula (see [

Finally, considering the hypothesis

Let

As in the proof of Theorem

Moreover, considering the relation

Therefore

As a consequence of the proof of Theorem

Let

The authors declare that they have no conflicts of interest.

This work is supported by National Natural Science Foundation of China (no. 11371076).